This excellent book should be read by the two audiences in the title: those interested in teaching mathematics and those interested in mathematicians who developed certain parts of mathematics in America during roughly 1890-1960. Indeed, anyone interested in the role of mathematics within American society during this period will also enjoy the biography and benefit from reading it. The book certainly would make a perfect gift or award for high-school or college students.

The protagonist, R. L. Moore (1882-1974), is known today for his method of teaching and for his contributions to mathematics, mostly topology. This well-researched biography, which is mostly drawn from written archival records and taped interviews with several important characters, does not paint Moore in a lily white setting. Instead, warts and endearing characteristics are portrayed equally in what amounts to a very fair portrait of a very fascinating figure. We examine the biography through the eyes of someone referred to only as "Mr. W" in the MAA film about the Moore Method titled Challenge in the Classroom.

Mr. W stated, "It was not simply the man, but also his ideas, but you can't separate the two." This assertion applies to the familiar refrain probably mouthed by all of us mathematics teachers at one point or another — mathematics is not a spectator sport. We know that students must become actively engaged with the subject in order to learn it. But how do we achieve this goal? The Moore Method is one way, and this book provides an excellent account of the method, not in just one small snippet, but by tracing Moore's development of the method during his sixty-four years of experience as a teacher. Yet the Moore Method is not appropriate for all teachers. Indeed, the biggest part of its original success probably was due to Moore himself, whose inordinate patience and insight into posing the right question at the right time liken him to a modern-day Socrates.

So, if few of us can duplicate the Moore Method, why then should mathematics teachers read the biography? Because, as Mr. W indicated, we cannot separate our own ideas from our personalities; in short, we each have to modify our teaching methods to meet our own situations. What Mr. W did not say, but which the book makes clear, is that success with any teaching method is a function of commitment to teaching. For Moore, that commitment was total, and he demanded a total commitment on the student's part in return.

Who is Mr. W? His full name is John M. Worrell, Jr. In the film Moore tells about how Worrell walked out of class in 1958-59 because he did not want to be told a certain concept. Two years later, in a subsequent class, Moore started to make a remark when Worrell, fearing that Moore was about to define that concept, seized the door knob, flung open the door, and rushed out of the room. Worrell had learned the vital lesson that he was empowered to think of the concept himself. He did not have to defer to external authority, any authority, whether it be eminent professor, revered classmate, or classical book. And isn't that the ultimate lesson we want our students to learn? Don't we tell them that there is no back-of-the-book in real life?

Worrell's example is one of the reasons why this book could be given to a high-school or college student, especially since the study of mathematics is not generally viewed today as particularly attractive. Worrell, in fact, was not born a mathematician. He entered the University of Texas as a pre-medical student intending to spend his career as a physician before encountering Moore in an algebra course in the summer of his freshman year. Moore's manner of teaching by asking questions intrigued Worrell, yet it was not until his last year on campus that he elected another course with him — calculus. It was here that Worrell received the full impact of the Moore Method. "It was very clear then that he wanted students to work independently (p. 267) ...he had an incredible degree of patience (p. 268)."

Worrell proceeded to earn his MD degree, but that is not the end of the story. He took additional courses with Moore that culminated with a PhD in 1961 and then a job with a research team at Sandia Laboratories on a project involving space medicine. It was there that Worrell solved a problem that played a crucial role in NASA being able to launch Mariner probes, even though his approach was at odds with the prevailing thought on the problem. He wrote that Moore's empowering of individuals "to work and think independently was applicable in industry and fields other than mathematics... I still feel to this day that Moore is one of the persons who helped get us to Mars (p. 301)."

Worrell serves as a representative of but one generation of American mathematicians introduced in this book. Earlier in Moore's career, one meets his mentors G. B. Halsted and E. H. Moore. Next comes a generation associated with the University of Chicago — L. E. Dickson, Gilbert Bliss, Oswald Veblen, and G. D. Birkhoff, the latter two extending coverage to Princeton and Harvard. Then we meet the first group of Moore PhD students — J. R. Kline at Penn, and then, at Texas, Raymond Wilder, Gordon Whyburn, and Burton Jones. They are followed by Gail Young, R H Bing, Edwin Moise, Richard Anderson, and Mary Ellen Rudin. You could write quite a history of mathematics in America just concentrating on these household names, but they all received their degrees before 1950, when Moore virtually stopped publishing and turned full attention to teaching. In the 50s he produced the likes of Bill Mahavier and Steve Armentrout before John Worrell stumbled into his algebra class. The biography is laced with photos of almost all of these figures and many more.

Clearly R. L. Moore is one of the towering figures in 20th-century mathematics, measured either by his contributions to topology or his training of researchers. This book traces his life from his early days (and even earlier in an appendix dealing with a genealogy that reveals surprising connections to both sides of the Civil War) through his forced retirement from teaching at age 86. Along the way the reader encounters various mathematical and non- mathematical developments. The final part deals with the ignoble handling of Moore's forced retirement; it reads like a historical novel. The reader will not be able to put it down until completing the last page, and even then will wish that a sequel was in store. Unfortunately, or fortunately, depending on your view, there can only be one R. L. Moore.

David E. Zitarelli is currently an associate professor of mathematics at Temple University. In alternate years he teaches a course in the general history of mathematics and in the history of mathematics in America, to both undergraduate and graduate students. He can be reached at david.zitarelli@temple.edu. His home page contains a selection of papers written by undergraduate students in his most recent class on the history of mathematics in America.